Physics · Oscillations · NEET
No. Every SHM is oscillatory, but not every oscillation is SHM. A ball bouncing between two walls oscillates, but its acceleration is not proportional to displacement. SHM needs the extra condition a = -w^2 x. So SHM is a special, cleaner type of oscillation.
The restoring force must obey F = -kx. The minus sign means the force always points toward the mean position. 'Proportional to x' means the force grows in a straight line as you move away. Both parts must hold. From F = ma you then get a = -(k/m) x = -w^2 x.
The minus sign is the whole idea of a restoring force. It says: whichever side of the mean position you are on, the force pushes you back. Without the minus sign the body would run away from the center, not oscillate. So sign opposite to displacement is a strict condition.
No. If a = +w^2 x (same sign as x), the motion is unstable and the body accelerates away, not back. SHM strictly needs a and x to have opposite signs, giving a = -w^2 x. Same-sign proportionality is not SHM.
Only for small angles. For a pendulum the restoring torque goes as sin(theta). SHM needs proportional-to-displacement, so we use the approximation sin(theta) is about theta, true only for small angles (roughly under 10 degrees). For large swings sin(theta) is not proportional to theta, so it is oscillatory but not exact SHM.
Write acceleration in terms of displacement. If a = -(positive constant) times x, it is SHM and that constant equals w^2. If there is a constant added, like a = -w^2 x + c, shift the origin to the new mean position and it is still SHM about that point. Any other power of x (like x^2 or 1/x) is not SHM.
The phase difference between displacement and acceleration of a particle in simple harmonic motion is
Try the real previous-year questions from this chapter — each with the answer and a full solution.
One: the restoring force is proportional to displacement and directed toward the mean position (F = -kx). Two: as a result, acceleration follows a = -w^2 x. If both hold, the motion is SHM.
For a spring-mass system w^2 = k/m, so w = square root of (k/m). In general w^2 is the positive constant of proportionality between acceleration and displacement. Its square root gives the angular frequency, and time period T = 2 pi / w.
Yes. A constant just shifts the mean position. Rewrite as a = -w^2 (x - c/w^2). About the new center the motion is still SHM with the same w. This is common with gravity in vertical spring systems.
Uniform circular motion is not SHM, but its projection on any one diameter is exactly SHM. That projection obeys a = -w^2 x, which is why SHM is often taught as the shadow of circular motion.
Almost every formula in the chapter (velocity, acceleration, energy, time period of spring and pendulum) is built on the single condition a = -w^2 x. Understanding this condition first makes the whole chapter fall into place.
